Kybernetika
Evžen Kindler
Mathematical theory of free rhythm
Kybernetika, Vol. 11 (1975), No. 3, (223)--233
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KYBERNETIKA —VOLUME 11 (1975). NUMBER 3
Mathematical Theory of Free Rhythm
EVZEN KlNDLER
A context free grammar describing synthesis of a phrase in free rhythm is presented. Free
rhythm is that used in ancient proses and in music of early middle age in Europe and in byzantine
influenced Asia. The grammar illustrates how modern means for communication analysis can
exactly describe a matter which has got no corresponding expressing means in natural languages.
1. INTRODUCTION
Free rhythm as a certain system of time organization in artistical creations has
been recognized in the modern time since about 100 years. After several years of
initial development of its research certain categories concerning it have been fixed
so that they could get a common spiritual property at least for specialists in corresponding scientific branches and that they have not been essentially modified by the
following research of free rhythm. The categories have been described e.g. in [1] or
[2] but in the greatest detail they have been presented in [3].
Free rhythm is not an absence of rhythm but a certain mode of ordering in time,
which is different from the rhytmical rules known in classical poetry or music of the
present millenium. It formed a basis of so called plainsong or plainchant, which was
a unisone mode of singing general in both the Roman imperiums (West Roman and
East one) since their separate existing and which was transferred into "barbarian"
countries as a cultural heredity, where it lived almost a thousand years. In high
middle age it began to be in decadence and with the end of middle age it was completely forgotten.
By modern scientists, the free rhythm system has been recognized first of all in the
oldest musical compositions based onlatine language texts, called milanese or ambrosian chant, and in the musical creation that followed: visigothic (or mozarabic) chant
and namely gregorian one. The same rhythmical rules have been discovered also
in greek and latin e prosaic creation well known from ancient time. Since about 40
years, when the original byzantine music has been redechiffred from medieval manuscripts after several centuries of ignorance (see [4]) the same system of rhythmical
synthesis has been discovered also in music based on greek language, which rose and
developed paralelly with the mentioned "latine" music. The explosion of its research
and the richness of the results of this research implies a certain hope that the system
of free rhythm holds also in other branches of musical eventually poetical creation
of the early middle ages in Armenia, Syria, Egypt and Aethiopia, but one must wait
until a sufficient number of compositions is authentically read.
The modern specialists have a certain terminology describing the rules of the free
rhythm system, but the understanding is given by a certain practice in realizing the
mentioned music: we can state that nowadays the musicologists, together with some
other specialists in philology and aesthetics have understood something which was
evident for the ancients but for which there are no suitable means in natural languages
so that they could transfer an objective information about free rhythm outside the
small group of specialists. In the present paper, the author tries to do a first step to
prepare some essential properties of free rhythm so that they would be independent
on musical sentiment and thus certain semiotical aspects of them could be communicated as notions of a formal theory. The step uses the phrase-structure-grammar
mechanism, well known from mathematical linguistics (see e.g. [5]). The following
part contains a description of the phrase-structure-grammar production rules of free
rhythm: any exact description is always preceded by a mention of the corresponding
property known from music.
2. DESCRIPTION O F THE SYSTEM
2.1. Elementary rhythm
In the free rhythm system we consider elementary time values. In natural languages
they are considered as short syllabes while in music they are usually written as
quavers. In the present paper we use a letter D to represent an elementary time value.
The elementary time value cannot be divided but it can be repeated. Thus we can
write DD, DDD etc. Let us mention that DD is an equivalent for a long syllabe in
natural languages. In the mathematical theory of free rhythm we do not study real
duration of an elementary time value, as it is an attribute which is outside the same
theory: in latine chant, it is ordinarily permitted a slight ritardando or accelerando
of any elementary time valne, but it is not specified of which size it has to be; the
same principles have been transferred into the byzantine chant according to the school
of Wellesz (see [4]) but after new considerations it seems that it will be possible
to formulate more exact laws for any ritardando or accelerando. These laws seem to
be more rich in more oriental music, as e.g. in music of Armenia or Syria. But for
the present paper it is important that the synthesis of the elementary time values
does not depend on their duration in realization.
Elementary rhythm reflects the ordering of elementary time values at the lowest
level: they can form either elementary arses or elementary theses. An elementary
arsis is one elementary time value and it introduces one instance of the elementary
rhythm by a certain elevation of the flow of the elementary time values. As that
elevation has no exact relation to any physical property of the elementary time value
forming the arsis, we shall write it so that we put a letter H before the corresponding
D. Thus we obtain the first production rule of the grammar of the free rhythm:
(1)
AE-*HD.
An elementary thesis can be either one elementary time value or a pair of them; we
consider it as a certain relaxation of the flow of the elementary time values which
can take one or two elementary time values. Also that relaxation has no exact relation
to any physical property of the corresponding elementary time values so that we must
introduce a new symbol / for it. It corresponds to the symbol of vertical episeme
which is used in books prepared for practical realization in modern time (naturally,
there are a lot of identification where one should put a thesis according to the context
or to the graphical form of neumes, so that the episemes are not to be necessarily
in all cases; in our system, where the neumes are not used and where only context-free
rules are applied, we must write symbol I before any thesis). Therefore we get the
production rules for the elementary thesis:
(2)
rE -» ID ,
(3)
TE -» IDD .
The elementary rhythm is a pair where after an elementary arsis an elementary
thesis follows. It can be reflected in the grammar by the following rule:
(4)
RE - AETE •
From the view point of elementary level any instance of a musical or linguistical
form we consider is a sequence of elementary rhythms.
2.2. Simple rhythm
Elementary rhythms form no efficient components for a higher rhythmical synthesis. Instead of them we must introduce compound time values which begin by an
elementary thesis which is followed by an elementary arsis; we shall represent them
225
by an auxiliary symbol Dc for which it is possible formulate immediately the following rule:
(5)
Dc -> TEAE .
Therefore any compound time value has a form of either IDHD or IDDHD. Using
the musical terminology, we can state that from the view point of basic level any
composition is a sequence of compound time values. They can be ordered in simple
rhythms so that some of them form simple arses while the other form simple theses:
(6)
As -> FDC ,
(7)
Ts -> GDC .
In these production rules the symbols F and G have a certain analogy with the symbols
H and J introduced for the elementary rhythm: F initiates a certain elevation which
is present during the whole following compound time value, while G initiates a certain
relaxation with a similar duration. Let us mention that these relaxations or elevations
can function against those of the elementary rhythm: any compound time value begins
with an elementary thesis.
The simple rhythm is formed by a pair of a simple arsis and a simple thesis:
(8)
Ks-AsTs.
2.3. Composed rhythm
We could be in a temptation to state that any instance of musical or linguistical
creation we consider is a sequence of simple rythms (such a statement would be an
analogy to that we have already formulated for the elementary and basic levels).
In fact it does not hold: the simple arses and theses can form chains which form
arses and theses on a higher level; moreover in these chains there can be present
also whole rhythms and thus one rhythm can be recursively composed of rhythms
on a lower level which need not be simple. This is an affair which cannot be expressed
by means of natural languages but which can be formalized by recursive means of
phrase structure grammars.
We introduce a notion of opened arsis, which has not been known by specialists
in music but which can serve us as a useful auxiliary mean:
(9)
(10)
(U)
A0 -> AP ,
A0^AA0,
Ao -• RA0 •
These rules tell that an opened arsis is a sequence of arses (rule (10)) and rhythms
(rule (11)) which must be concluded by an arsis (not by a rhythm — rule (9)), after
which a symbol P takes place. That symbol corresponds to the sign of petaste well
known in byzantine notation, where it indicates - similarly as P in our grammar —
the place of the greatest elevation effect in a rhythm. Let us mentioned that rule (9)
tells that a chain forming an arsis cannot be concluded by a thesis.
Similarly we introduce a notion of opened thesis:
(12)
To -
GT,
(13)
To -
(14)
To -> TQR .
T0T,
These rules have symetrical properties if compared with those for the opened arsis:
any opened thesis is a sequence of theses (rule (13)) and rhythms (rule (14)) which
begins with symbol G followed by a thesis (rule (12)). The symbol G has been introduced in rule (7) for simple rhythm; here it has a similar semantics, i.e. it introduces
a certain relaxation which durates during the whole chain forming the thesis. Let us
mention that rule (12) exclude any case that after G an arsis would be present.
The last 6 rules have used symbols A, T and R which correspond to the notions
of arsis, thesis and rhythm which are understood in their complete extent. We must
present rules for their forming. Those rules, presented further, expresses that any
rhythm is an ordered pair of arsis and thesis and arses and theses rise from opened
arses and theses respectively so that we "close" them. The opened arses must be closed
at their left hand sides by a symbol F: it has been introduced in rule (6) and its
semantics is generalized for the whole chain forming the arsis, similarly as it has been
alreday mentioned for the symbol G in case of thesis. Rule (15) expresses that closing:
(15)
A - FA0 .
Similarly any thesis is obtained from an opened thesis by closing it at its right hand
side by a symbol M. It has not been introduced; it means the point of the greatest
size of relaxation and we have used a letter M for it because its syntactical function
ressembles that of membre (or similarly of incisus or phrase) pause (see e.g. [2],
p. 90 and p. 108-116).
(16)
T->T0M.
Rules (15) and (16) must be completed by rules which would allow to form the most
simple arses and theses: they are simple ones closed similarly as the other arses and
theses.
(17)
A -> ASP ,
(18)
T -* TSM .
228
The last rule defines the forming of rhythm:
(19)
R -* AT.
It tells that any rhythm is an ordered pair of arsis and thesis. Let us mention that the
notion of rhythm which has been just introduced is not a direct generalization of
elementary rhythm or simple one. The most simple rhythm defined by rule (19) is
the following one:
FIDHDPGIDHDM
Its arsis FID HDP and thesis GIDHDM have risen from simple arsis FIDHD and
a simple thesis GIDHD by closing (adding P and M) respectively. The corresponding
simple arsis and thesis can form a corresponding simple rhythm
FIDHDGIDHD
which is not a rhythm according to the rule (19).
2.4. Phrases
Forming of more composed rhythms according to the rules of the last section
can be repeated until forming a phrase. It is a rhythm which is sufficiently rich from
the point of view of the author intention but which cannot form a component in
forming more complicated rhythm (according to rules (11) and (14)) because such
rhythms would be too long than they could be taken by any human as rhythmical
entiers. Thus we can formulate a rule defining the notion of phrase which is identified
byK:
(20)
K -> R .
The rule must be completed by another one so that it would be possible to allow
also elementary rhythm to form a phrase, because we meet with such instances
of phrases in plainsong. Thus we formulate the last rule
(21)
K -> RE .
One can state that any composition of the array which we recognize is a sequence
of phrases. Any phrase is a result of the highest rhythmical synthesis of the elementary
time values while the ordering of phrases is no more an affair of rhythm but of that
which is commonly called musical or stylistical form. It does not concern the interests and the method of the present paper.
Therefore we can conclude our derivation of the rules of the free rhythm system
and close their system into a phrase structure grammar:
Grammar of free rhythm is an ordered 4-tuplet T
of terminal symbols which are namely D, I, F, G,
of auxiliary symbols which are namely AE, TE, R E ,
and K, where K e V. is the initial symbol and S
to (21).
= (V, Vu K, S) where Vis a set
H, P and M, where Vt is a set
Dc, As, Ts, R s , A0, T 0 , A, T, R
is a set of production rules (1)
3. SIMPLIFICATION OF THE GRAMMAR
We have tried to transform all the rules known from the non-mathematical theory
of free rhythm into production ones and for the legibility we have let them enter
into the set S of rules. We can see that for any ordinary composition of a phrase
the rule (8) is not necessary (it does not take place in the right hand side of any
production rule) and thus we can omit it from S. Then we are able to omit symbol
R s from Vi.
Another simplification can be made by omitting rule (21). It can be admitted
because any elementary rhythm forming a phrase can be completed to a simple
rhythm (and then according to rules (17) and (18) to a rhythm) so that we put an
elementary time value before that which forms its arsis, and we eventually put
another elementary time value after that which forms its thesis (in case that the
elementary thesis consists of one elementary time value — such a case takes place
only in byzantine music while in the latine one it has not been met). So e.g. an elementary rhythm HDIDD which forms a phrase (and which is thus not used as a component
of a rhythm in a higher level) can be transforfed into FIDHDPGIDHDM
which
satisfies rule (19) and which really differs from the original one only by the first
occurence of D which can be considered as a quaver pause, i.e. which is not realized
but silently felt (such a practice has been introduced also in many modern transcriptions of plainchant).
After omitting rule (21) from S we can omit also rules (l) to (4) from the same set
and the corresponding symbols RE, AE and TE from set Vx. Moreover we can omit
also terminal symbols H and / from set V because their occurence (and thus the
corresponding semantical function) can be determined from the context: if we omit
all the occurences of both symbols from any chain of terminal symbols forming
a phrase we can return them simply according to the following procedure: we read
the chain from left to right; if we meet a configuration DD we replace it immediately
by a configuration IDHD; if the following symbol is D we let it be present in the
chain; in any case we repeat the test for occurence of DD at the right of the put
configuration IDHD or at the right of an eventual repeated occurence of D. The
algorithm is exactly described at the end of the present paper.
If we omit H and / from V we must correspondingly modify rule (5) replacing
it by 2 new rules:
(5a)
D c -> DD,
(5b)
Dc -y DDD .
They tell that any compound time value is either a pair or a triplet of elementary
time values.
Another simplification is caused by the fact that after introducing the notions
of the simple rhythm in 2.2 we had to formulate special production rules for further
synthesis of them (rules (17) and (18)); it is possible to express another set of production rules so that they would be common for simple arses and theses (closed naturally
by all necessary symbols F, P, G and M) and for those composed at a higher level
according to the rules of 2.3. Therefore we omit rules (6), (7), (17) and (18) and we add
two new rules; they should logically complete the triplets of rules (9) to (ll) and (12)
to (14) respectively:
(22)
Ao -> DCP ,
(23)
To -> GDC .
They introduce partial cases of opened arses and theses, which could be called
opened simple arsis and thesis and which can be closed by the same way as the other
types of opened arses and theses using rules (15) and (16).
The last simplification consists in the fact that after omitting rule (21) there is only
one rule with K, namely rule (20). It can be omitted, if R becames the initial symbol
of the grammar. It reflects the fact that every rhythm can be a phrase.
Now we can present the modified grammar T* = (V*, V*, R, S*) which is more
simple than T but which offer an equivalent rhythmical synthesis: V* =
= (D, F, G, P, M), V* = (Dc, Ao, To, A, T R) and S* is the set of the following
production rules:
(5a)
Dc^
(5b)
Dc -* DDD ,
(22)
Ao -> DCP ,
(9)
DD,
A0-+AP,
(10)
Ao -* AAo ,
(11)
Ao->RA0,
(23)
To -> GDC ,
(12)
TQ^GT,
(13)
To -
(14)
To -»
(15)
A
(16)
T -> TQM ,
(19)
R
T0T,
T0R,
-+TAo,
-+AT.
4. EXTENSION O F THE GRAMMAR
The presented grammars have been constructed according to the theoretical
informations about the free rhythm system. Sometimes we meet examples detailly
prepared by the same theoreticians, which follow more general rules.. It would be
useful to extend the grammar so that it could reflect also those rules, which have not
been explicitely formulated but which have been implicitely felt when realizing the
considered type of music.
As an illustration we present an extension which rises from the example presented
in [6], page XIV. The example contains a short phrase with prepared arses and
theses of all levels. They have been mapped by usual signs for crescendo (arses) and
decrescendo (theses), and the points where any thesis is joined with the arsis of the
corresponding rhythm are marked by a sign of accent (it corresponds to our symbol P
or G). The example is presented in Fig. 1 (the text of the example, the signs concerning
Fig. 1.
elementary rhythm and some historical names of certain rhythms have been omitted
from it as they have no relation to the matter of the present paper). We can see
that there are two places in the example which are not consistent with the theory:
they have been identified by letters A and B under the notation; in both places
a rhythm begins which has "empty" arsis. Such an affair has not been admitted by the
theory but it can be simply felt by musicians as a certain discontinuity of the rhythmic
231
232
flow. It can be simply introduced into the presented mathematical theory: the set S*
of the production rules of the grammar T* is to be enlarged by one rule, namely
(15a)
A -» FP .
Then the presented example can be simply transformed to the following string:
FDDPGGGGDDMGDDMMFPGGDDMFDDPGDDMGDDDMMMFFPGDD
-
MFDDDPPGGDDMGDDMMM
The symbols D correspond to the quavers (elementary time values) of the original
notation; let us mention that any crotchet must be replaced by 2 instances of D.
5. CONCLUSIONS
We have presented certain tools which can exactly reflect rhythmical laws of ancient music and of related linguistical creations. We do not intend them to replace
usual notation in order to enable more exact definition of any relalization, because
the texts generated by the corresponding production rules are very complicated, as
we could see. The mathematical theory can serve more for research than for practice.
Nevertheless one can try to simplify the presented rules in order to get more legible
sequences of terminal symbols. The other way to develope the presented theory is
to include other rules of free rhythm which have been implicitely felt but which have
not been built into non-mathematical theories of free rhythm: such considerations
would serve for direct enriching of musical science.
We present an algorithm which completes texts generated by grammar E* by
symbols H and / and so it transforms them to texts generated by grammar I \ In other
words, the algorithm determines the elementary rhythm according to the rhythm
of a higher level. The algorithm is written as a program in ALGOL 68 (see [7]);
it reads a sequence of letters (a text generated by grammar F*) which is concluded
by a "sentinel", which is a symbol which is not included in Fand which informs that
the input sequence of symbols is finished. We use letter Z for it. The algorithm
generates another sequence of symbols (a text generated by grammar r corresponding to the read text).
begin char x; bool p : = false;
L
:read(x);
\lx 4= "D" then g o t o Affi;
if p then p : = false; print ("IDHD")
go to L;
M:iip
thenp : = false; print ("D")
else p : = true f i;
fi;
print (x); if x H= " Z " then go to L f i
end;
(Received July 30, 1974.)
REFERENCES
[1] G. Suňol: Text book of gregorian chant (in English; also in French, German, Spanish and
Italian). Desclée, Tournai 1930.
[2] P. Ferretti: Principii teoritici e pratici di Canto gregoriano. Rome 1933.
[3] A. Mocquereau: Le nombre musical grégorien ou rhythmique grégorienne. Tournai 1908
(Tom I), 1922 (Tom II).
[4] H. J. W. Tillyard: Handbook of the middle byzantine musical notation. Munksgaard, Co­
penhagen 1935.
[5] A. Salomaa: Formal languages. Academic Press, N. York—London 1973.
[6] Liber usualis missae et officii. Desclée, Tournai 1964.
[7] C. H. Lindsey, S. G. van der Meulen: Informal introduction to ALGOL 68. North-Holland,
Amsterdam—London 1971.
PhDr. RNDr. Evžen Kindler, CSc; Katedra matematické informatiky,
Matematicko-fyzikální
fakulta UK (Department of Mathematical Informatics, Faculty of Mathematics and Physics,
Charles University), Malostranské nám. 2/25, J18 00 Praha 1. Czechoslovakia.